Finite Group Databasebeta

← C23×#14583827 #36368362

Subgroups — C23×#14583827 #36368362

Order 22144.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14583827C2156yesmaximal, direct product factor
2C23×D692C213yesmaximal, complemented
3C23×C4×D173C213yesmaximal, complemented
4C23×Q8×C173C211yesmaximal, characteristic, complemented
5C24×D173C2213yes—
6C2×#14583827C221448yesdirect product factor
7C22×D692C22184yescomplemented
8C23×Dic173C2211yescharacteristic
9C22×C4×D173C22184yescomplemented
10C22×Q8×C173C22128yescomplemented
11C23×C4×C173C2213yes—
12C23×D173—224no—
13C23×D173C23142yes—
14#14583827C231512yesdirect product factor
15C2×D692C231336yescomplemented
16C22×Dic173C23114yes—
17C2×C4×D173C231336yescomplemented
18C24×C173C2311yescharacteristic
19C2×Q8×C173C231112yescomplemented
20C22×C4×C173C23142yes—
21C22×D173—2168no—
22C22×D173C24184yes—
23D692C241192yescomplemented
24C2×Dic173C24128yes—
25C4×D173C241192yescomplemented
26C23×C173#4118yescomplemented
27C23×C173C2417yes—
28Q8×C173C24164yescomplemented
29C2×C4×C173C24184yes—
30C22×C173C2×#41128yescomplemented
31C22×C173C2517yes—
32C2×D173—2168no—
33C2×D173C25124yes—
34C4×C173C25124yes—
35Dic173C2518yes—
36C2×C173C22×#41114yescomplemented
37C2×C173C2611yescharacteristic
38D173—224no—
39C173C23×#4111yescharacteristic, complemented, Sylow
40C23×#41—1731nomaximal, Sylow
41C22×#41—17356no—
42C23×Q8D17311yescharacteristic, complemented
43C23×D4—1733no—
44C24×C4—1733no—
45C25—1733no—
46C2×#41—173448no—
47C22×Q8C2×D173128yescomplemented
48C22×D4—17384no—
49C23×C4—1731no—
50C23×C4—17384no—
51C23×C4C2×D17313yes—
52C24—34624no—
53C24—17342no—
54C24C22×D17311yescharacteristic
55#41—173512no—
56C2×Q8C22×D1731112yescomplemented
57C2×D4—173336no—
58C22×C4—17314no—
59C22×C4—173336no—
60C22×C4C22×D173142yes—
61C23—346168no—
62C23—17384no—
63C23#1458382718yesdirect product factor
64C23C23×D17317yes—
65Q8C23×D173164yescomplemented
66D4—173192no—
67C2×C4—17328no—
68C2×C4—173192no—
69C2×C4C23×D173184yes—
70C22—346168no—
71C22—17324no—
72C22C2×#14583827128yesdirect product factor
73C22C24×D17317yes—
74C4—1738no—
75C4C24×D173124yes—
76C2—34624no—
77C2C22×#14583827114yesdirect product factor
78C2C25×D17311yescharacteristic
subgroupG/H
centerC24C22×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC23×Q8×C173C2
socleC24×C173C23
hypercenterC23×Q8D173
hypocenterC173C23×#41
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C24 ◁ C23×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#14583827C2direct (N × H)
2. C23×D692C2semidirect
3. C23×C4×D173C2semidirect
4. C23×Q8×C173C2semidirect
6. C2×#14583827C22direct (N × H)
7. C22×D692C22semidirect
9. C22×C4×D173C22semidirect
10. C22×Q8×C173C22semidirect
14. #14583827C23direct (N × H)
15. C2×D692C23semidirect
17. C2×C4×D173C23semidirect
19. C2×Q8×C173C23semidirect
23. D692C24semidirect
25. C4×D173C24semidirect
26. C23×C173#41semidirect
28. Q8×C173C24semidirect
30. C22×C173C2×#41semidirect
36. C2×C173C22×#41semidirect
39. C173C23×#41semidirect
42. C23×Q8D173semidirect
47. C22×Q8C2×D173semidirect
56. C2×Q8C22×D173semidirect
63. C23#14583827direct (N × H)
65. Q8C23×D173semidirect
72. C22C2×#14583827direct (N × H)
77. C2C22×#14583827direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145838272
C23×D6922
C23×C4×D1732
C23×Q8×C1732
C23×#41173